179 research outputs found
Exponential Asymptotics in a Singular Limit for -Level Scattering Systems
The singular limit \eps\ra 0 of the -matrix associated with the equation
i\eps d\psi(t)/dt=H(t)\psi(t) is considered, where the analytic generator
H(t)\in M_n(\C) is such that its spectrum is real and non-degenerate for all
. Sufficient conditions allowing to compute asymptotic formulas for the
exponentially small off-diagonal elements of the -matrix as \eps\ra 0 are
explicited and a wide class of generators for which these conditions are
verified is defined. These generators are obtained by means of generators whose
spectrum exhibits eigenvalue crossings which are perturbed in such a way that
these crossings turn to avoided crossings. The exponentially small asymptotic
formulas which are derived are shown to be valid up to exponentially small
relative error, by means of a joint application of the complex WKB method
together with superasymptotic renormalization. The application of these results
to the study of quantum adiabatic transitions in the time dependent
Schr\"odinger equation and of the semiclassical scattering properties of the
multichannel stationary Schr\"odinger equation closes this paper. The results
presented here are a generalization to -level systems, , of results
previously known for -level systems only.Comment: 35 pages, Late
Density of States and Thouless Formula for Random Unitary Band Matrices
We study the density of states measure for some class of random unitary band
matrices and prove a Thouless formula relating it to the associated Lyapunov
exponent. This class of random matrices appears in the study of the dynamical
stability of certain quantum systems and can also be considered as a unitary
version of the Anderson model. We further determine the support of the density
of states measure and provide a condition ensuring it possesses an analytic
density
An Adiabatic Theorem for Singularly Perturbed Hamiltonians
The adiabatic approximation in quantum mechanics is considered in the case
where the self-adjoint hamiltonian , satisfying the usual spectral gap
assumption in this context, is perturbed by a term of the form . Here is the adiabaticity parameter and is a
self-adjoint operator defined on a smaller domain than the domain of .
Thus the total hamiltonian does not necessarily
satisfy the gap assumption, . It is shown that an
adiabatic theorem can be proven in this situation under reasonnable hypotheses.
The problem considered can also be viewed as the study of a time-dependent
system coupled to a time-dependent perturbation, in the limit of large coupling
constant.Comment: 17 pages, LaTe
Spectral Properties of Non-Unitary Band Matrices
We consider families of random non-unitary contraction operators defined as
deformations of CMV matrices which appear naturally in the study of random
quantum walks on trees or lattices. We establish several deterministic and
almost sure results about the location and nature of the spectrum of such
non-normal operators as a function of their parameters. We relate these results
to the analysis of certain random quantum walks, the dynamics of which can be
studied by means of iterates of such random non-unitary contraction operators.Comment: updated version, to appear in Annales Henri Poincar
Fractal Weyl Law for Open Chaotic Maps
This contribution summarizes our work with M.Zworski on open quantum open
chaoticmaps (math-ph/0505034). For a simple chaotic scattering system (the open
quantum baker's map), we compute the "long-living resonances" in the
semiclassical r\'{e}gime, and show that they satisfy a fractal Weyl law. We can
prove this fractal law in the case of a modified model.Comment: Contribution to the Proceedings of the conference QMath9,
Mathematical Physics of Quantum Mechanics, September 12th-16th 2004, Giens,
Franc
Dynamical Localization of Quantum Walks in Random Environments
The dynamics of a one dimensional quantum walker on the lattice with two
internal degrees of freedom, the coin states, is considered. The discrete time
unitary dynamics is determined by the repeated action of a coin operator in
U(2) on the internal degrees of freedom followed by a one step shift to the
right or left, conditioned on the state of the coin. For a fixed coin operator,
the dynamics is known to be ballistic. We prove that when the coin operator
depends on the position of the walker and is given by a certain i.i.d. random
process, the phenomenon of Anderson localization takes place in its dynamical
form. When the coin operator depends on the time variable only and is
determined by an i.i.d. random process, the averaged motion is known to be
diffusive and we compute the diffusion constants for all moments of the
position
Spectral Properties of Quantum Walks on Rooted Binary Trees
We define coined Quantum Walks on the infinite rooted binary tree given by
unitary operators on an associated infinite dimensional Hilbert space,
depending on a unitary coin matrix , and study their spectral
properties. For circulant unitary coin matrices , we derive an equation for
the Carath\'eodory function associated to the spectral measure of a cyclic
vector for . This allows us to show that for all circulant unitary coin
matrices, the spectrum of the Quantum Walk has no singular continuous
component. Furthermore, for coin matrices which are orthogonal circulant
matrices, we show that the spectrum of the Quantum Walk is absolutely
continuous, except for four coin matrices for which the spectrum of is
pure point
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